# Holomorphy of Adjoint L-functions for $GL(n):$ $n\leq 4$

## TL;DR

This paper proves the entire nature of adjoint L-functions for cuspidal representations of GL(3) and GL(4) over any global field, including twisted cases, advancing understanding of their analytic properties.

## Contribution

It establishes the entireness of complete adjoint L-functions for GL(3) and GL(4), extending previous results to all cuspidal representations over arbitrary global fields.

## Key findings

- Proves entireness of adjoint L-functions for GL(3) and GL(4)
- Includes twisted cases in the analysis
- Applies to any cuspidal representation over global fields

## Abstract

We show entireness of complete adjoint L-functions associated to \textbf{any} cuspidal representations of $\GL(3)$ or $\GL(4)$ over an arbitrary global field. Twisted cases are also investigated.